Changing a humidifier filter is actually not a complicated task when it comes to most standard units. The most important thing to remember is that manufacturer instructions should trump any general guide such as this one. That being said, here is a very high level look at how to install a new replacement humidifier filter. Since we are going into the unit, it is also a good time to do some very basic cleaning.
The first part of the humidifier you have to remove is the water tank. Dispose of any sitting water and rinse the tank. Then fill it partially and add a few drops of bleach to disinfect any possible bacteria growth that might have accumulated with the sitting water. Then remove the motor housing which is the biggest piece covering most of the unit. With the housing removed, this should reveal the old filter which should easily slide or lift off and out. Clean the base and around the humidifier before installing the new filter.
Some humidifier filters are reusable meaning that they can be wasted and reused instead of replacing it. In many cases, it makes more sense to replace the filter since it is a sure way of knowing the filter is completely clean. In the event you do decide to reuse the filter, and this is only if there are specific instructions that came with the unit stating the filter is reusable, it can be reinstalled wet and there is no reason to wring or squeeze the filter dry. That only puts it at risk of tearing or otherwise being damaged making it an ineffective filter.
Putting everything back together should be as easy as taking things apart. One precaution is that you do not take the unit apart more than it is necessary to replace the filter. Doing so will most likely void any warranty that might have come with the unit so check the specific details pertaining to your products warranty before proceeding.